BRST-invariant Pauli-Villars regularization of QCD
aa r X i v : . [ h e p - t h ] J un BRST-invariant Pauli–Villars regularization of QCD
Sophia S. Chabysheva and John R. Hiller
Department of Physics and AstronomyUniversity of Minnesota-DuluthDuluth, Minnesota 55812 (Dated: July 7, 2018)We extend the QCD Lagrangian to include Pauli-Villars (PV) gluons, quarks, and ghosts in sucha way as to retain BRST invariance in an arbitrary covariant gauge. The extended Lagrangiancan provide a starting point for nonperturbative calculations in QCD, particularly with light-fronttechniques, and the methods used to construct it may be useful for perturbative calculations intheories where dimensional regularization is not viable. The regularization is arranged by havingall interaction terms in the Lagrangian be couplings between null fields, specific combinations ofpositive and negative-metric PV fields. The construction is done in steps, beginning with a gauge-invariant Lagrangian with massless PV gluons and degenerate-mass PV quarks. Auxiliary scalarsare introduced to give mass to the PV gluons, and break the mass degeneracy of the PV quarks,following a method due to Stueckelberg. Gauge fixing terms for the gluon fields are a part of thisconstruction. The ghost terms are then obtained and shown to provide the BRST invariance. Alack of dependence on the gauge parameter can be checked by the calculation of physical quantitiesfor a range of values of the parameter.
The principal goal of hadronic physics is the nonper-turbative solution of quantum chromodynamics (QCD),in order to compute properties of hadrons as bound statesof quarks and gluons. Lattice gauge theory [1] is one ap-proach for doing this; it has met with considerable suc-cess, but is limited by its reliance on a Euclidean frame-work and by other difficulties, such as fermion doublingand a large pion mass. Dyson–Schwinger methods [2] arealso useful but also Euclidean. The truncated conformalspace approach [3] shows promise but is in its infancy.An alternative, which retains a Minkowski frameworkand the intuitive notion of wave functions, is the light-front Hamiltonian approach [4, 5]. Light-front methodshave been successfully applied to many gauge field the-ories, including quantum electrodynamics [6–8], QCDin two dimensions [9], and supersymmetric Yang–Millstheories [10]; however, with respect to four-dimensionalQCD, this approach has always lacked a consistent reg-ularization that can be used nonperturbatively.The standard regularization used for non-Abelian the-ories such as QCD is dimensional regularization [11]. Un-fortunately, this method is inherently perturbative, rely-ing as it does on the analysis of the singularity structureof Feynman diagrams and the modification of the asso-ciated integrations. All of this is buried deep within anonperturbative calculation, where such separations andmodifications are difficult if not impossible to perform.A regularization that can be introduced at the La-grangian or Hamiltonian level is much to be preferred,and there has been just such an approach for Abeliantheories, developed in the context of light-front Hamil-tonian methods [6, 12, 13]. Massive Pauli–Villars (PV) There has also been work on construction of a light-front regu- fields are introduced to the Lagrangian with couplingsarranged to provide the cancellations necessary for reg-ularization via subtractions of negative-metric PV con-tributions, and the Hamiltonian derived in some fixedgauge.The cancellations are obtained by constructing all in-teractions from combinations of positive and negative-metric fields that are null in the following sense. Let the(light-front) mode expansions for the gluon and quarkfields be written as A akµ ( x ) = Z dq √ π X λ e ( λ ) µ (cid:2) a akλ ( q ) e − iq · x (1)+ a † akλ ( q ) e iq · x i ,ψ i ( x ) = Z dq p π q + X as (cid:2) u ais ( q ) b ais ( q ) e − iq · x (2)+ v ais ( q ) d † ais ( q ) e iq · x i , with dk = dk + d k ⊥ the light-front volume element [4], e ( λ ) µ a set of polarization vectors, a † akλ the gluon creationoperator with color index a , PV index k , and polarization λ , and b † ais the quark creation operator with color index a , PV index i , and spin s . The quark flavor index is sup-pressed, as it will be throughout. The physical fields aredenoted by a PV index of zero. The (anti)commutationrelations for the creation operators are[ a akλ ( q ) , a † blλ ′ ( q ′ )] = r k ǫ λ δ ab δ kl δ λλ ′ δ ( q − q ′ ) , (3) { b ais ( q ) , b † bjs ′ ( q ′ ) } = s i δ ab δ ij δ ss ′ δ ( q − q ′ ) . (4) larization that is perturbatively equivalent to covariant regular-ization [14]. Here r k = ± s i = ± k and i , respec-tively. In addition, the gluon field has a polarization de-pendent metric signature specified by ǫ λ ≡ ( − , , , A µa ≡ X k ξ k A µak , ψ ≡ X i β i ψ i , (5)where the ξ k and β k are coupling coefficients constrainedto satisfy X k r k ξ k = 0 , X i s i β i = 0 . (6)Obviously, at least one PV gluon and one PV quark musthave a negative metric. Typically one would take r k =( − k and s i = ( − i .Interaction terms, such as g ¯ ψγ µ T a A aµ ψ , are then builtfrom null combinations, with r = s = ξ = β = 1 forthe physical field. Here T a is a matrix representing theLie algebra, and repeated color indices are summed. Anyloop must then contain the sum P ik s i β i β ′ i r k ξ k ξ ′ k , withthe primed coefficients coming from the second vertex,which may or may not be the same as the first. The met-ric signatures come from contraction of the creation andannihilation operators associated with the internal linesof the loop. If the two vertices are the same, the nullconstraints immediately provide two subtractions; if notthe same, the different null combinations from the differ-ent vertices must be mutually null, that is P i s i β i β ′ i = 0and P k r k ξ k ξ ′ k = 0, to again provide two subtractions.This approach has been used to study QED in an arbi-trary covariant gauge, including study of the dependenceof physical quantities on the gauge parameter [8]. Thecouplings induce currents that change the PV index ofthe field, including mixing with the physical field, whichbreaks gauge invariance; however, in the limit that thePV fields are removed, gauge invariance is restored.The extension to non-Abelian theories is, of course,problematic. Unlike Abelian theories, where a massivevector boson is known to be renormalizable [15], the proofof renormalizability for Yang–Mills theories is best ap-proached in terms of BRST invariance [16], which wouldappear to require an underlying gauge invariance, whichin turn requires massless vector bosons. The well-knownexception is the use of spontaneous symmetry breaking toprovide mass to gauge bosons without destroying gaugeinvariance at the Lagrangian level, which is a hint as tohow one might proceed. Another difficulty is that, evenfor massless PV gluons and mass-degenerate PV quarks,the null couplings that mix fields do break gauge invari-ance.We have resolved these difficulties. The giving ofmass to the PV gluons and the lifting of the PV-quarkmasses is accomplished by a non-Abelian generalizationof Stueckelberg’s mechanism [17, 18], which associates an auxiliary real scalar with each gluon field, simulta-neously giving mass to the gluon and the scalar whilealso fixing the gauge. The breaking of gauge invarianceby the field-mixing interactions is eliminated by general-izing the definition of the gauge transformation to alsoinclude field mixing; the original gauge transformation isrecovered in the limit that the PV fields are removed, bytaking their masses to infinity. The BRST invariance isrecovered for finite PV masses by inclusion of the appro-priate Faddeev–Popov [19] ghost terms required by thegauge-fixing terms.In case this sounds too good to be true, we must pointout that there are drawbacks to our approach. One is thepresence of nonlocal interactions for the ghost fields; thisis made necessary by the mechanism used to generate PVgluon masses. The other drawback is a proliferation ofPV fields, which will be a large computational burden.There are several constraints that the PV couplings mustsatisfy, and each constraint requires the addition of oneor two PV fields. The formulation given here can requireas many as four PV gluons, three PV quarks, five PVadjoint real scalars, and four PV ghosts and anti-ghosts.This is for each color and flavor, and therefore is a verylarge number.Oddly enough, the Higgs mechanism is not used. Wefound that the requirement of null couplings, to providethe regularization, causes two difficulties. One is in theHiggs sector itself, where the null φ interaction does notlead to a well-defined minimum to drive the breaking ofthe symmetry. The other is that gauge-invariant null cou-plings of the PV Higgs to the gluons requires that anysymmetry breaking leave as massless only a null com-bination of gluon fields, rather than produce a masslessphysical gluon and massive PV gluons.In what follows, we describe the construction in stages.The first is a Lagrangian for massless gluons and mass-degenerate quarks that is invariant with respect to a gen-eralized gauge transformation. Next, we introduce theauxiliary scalars and add terms that give mass to the PVgluons and fix the gauge. This is followed by specifica-tion of a term that lifts the mass degeneracy of the PVquarks. Finally, we determine the Faddeev–Popov ghostterm and state the BRST transformations of the fields.Our construction is kept slightly more general than QCD,in that we treat SU ( N ) Yang–Mills theory with funda-mental matter, with QCD being the N = 3 case.The gauge-invariant Lagrangian for massless PV glu-ons and mass-degenerate PV quarks is L = − X k r k F µνak F akµν + X i s i ¯ ψ i ( iγ µ ∂ µ − m ) ψ i + g X ijk β i β j ξ k ¯ ψ i γ µ T a A akµ ψ j . (7)The quark spinor field ψ i is a column vector with respectto color. The quarks (of a single flavor) are all of mass m . The gluon field tensor F µνak is computed from the fieldas F µνak = ∂ µ A νak − ∂ ν A µak − r k ξ k gf abc X lm ξ l ξ m A µbl A νcm . (8)The structure constants f abc specify the commutation re-lation [ T a , T b ] = if abc T c .The gauge transformations of the fields are A µak −→ A µak + ∂ µ Λ ak + r k ξ k gf abc Λ b A µc , (9) ψ i −→ ψ i + igs i β i T a Λ a ψ, (10)with Λ a ≡ P k ξ k Λ ak . The null combination ψ is thengauge invariant, and the null combination A µa is Abelian A µa −→ A µa + ∂ µ Λ a , (11)as is the associated field tensor F µνa = X k ξ k F µνak = ∂ µ A νa − ∂ ν A µa . (12)The Lagrangian L is gauge-invariant with respect tothese transformations.When expressed in terms of the null combinations, theLagrangian reduces to L = − X k r k ( ∂ µ A νak − ∂ ν A µak ) + gf abc ∂ µ A νa A bµ A cν + X i s i ¯ ψ i ( iγ µ ∂ µ − m ) ψ i + g ¯ ψγ µ T a A aµ ψ. (13)All interactions are between null combinations, andthe free part of the Lagrangian corresponds to termsfor massless gluon fields with metric signatures r k anddegenerate-mass quark fields with signatures s i . Thefour-gluon interaction that normally appears in the QCDLagrangian has disappeared, as a result of cancellationsbetween physical and PV gluons. It is restored betweenphysical gluons in the infinite-mass limit for PV gluonsas a contraction of two three-gluon vertices, with the con-traction being a PV gluon. This provides a mechanism,natural in the context of the present Lagrangian, for theoften-used trick of introducing an auxiliary field to reducethe four-gluon interaction to two three-gluon interactionsfor the convenience of color factors in perturbation the-ory [20].To give mass to the PV gluons, we use a non-Abelianextension of the Stueckelberg mechanism [17, 18]. Realadjoint scalar fields φ ak are introduced with a gaugetransformation of φ ak −→ φ ak + µ k Λ ak + µ k r k ξ k gf abc Z x dx ′ µ Λ b ( x ′ ) A µc ( x ′ ) . (14)The line integral is present in order that the gauge trans-formation of the derivative be ∂ µ φ ak −→ ∂ µ φ ak + µ k ∂ µ Λ ak + µ k r k ξ k gf abc Λ b A µc . (15) This then makes the combination µ k A µak − ∂ µ φ ak gaugeinvariant, which is used in the construction of the firstterm of an additional piece for the Lagrangian, writtenas L g = 12 X k r k ( µ k A µak − ∂ µ φ ak ) (16) − ζ X k r k (cid:18) ∂ µ A µak + µ k ζ φ ak (cid:19) . The second term is the gauge fixing term. However, whenthe two terms are combined, the cross terms sum to atotal divergence which can be neglected, leaving L g = 12 X k r k µ k ( A µak ) − ζ X k r k ( ∂ µ A µak ) (17)+ 12 X k r k (cid:20) ( ∂ µ φ ak ) − µ k ζ φ ak (cid:21) . This is immediately seen to provide a mass µ k and stan-dard gauge-fixing term for each gluon field, with gaugeparameter ζ , and a free Lagrangian for real scalars withmasses µ k / √ ζ and metric signatures r k . For the physicalgluon, the mass µ is to be taken to zero. For pure Yang–Mills theory, this can be done explicitly, by exclusion ofthe scalar fields φ a and replacement of the k = 0 termsin L g by a simple gauge-fixing term for the physical glu-ons: − ζ ( ∂ µ A µa ) . When quarks are included, the k = 0scalar is needed, and the mass µ must be taken to zeroas a limit; however, as will be seen, the scalar φ a doesnot couple to the physical fields for any value of µ .The mass degeneracy of the PV quarks can be re-moved by coupling the quarks to a null combination ofthe scalars φ ak , e φ a ≡ X k ξ k µ PV µ k φ ak , (18)made null by the additional constraint X k r k ξ k µ k = 0 . (19)Here µ PV ≡ max k µ k is the mass scale of the PV gluons.The gauge transformation of the combination is Abelian e φ a −→ e φ a + µ PV Λ a (20)and provides the necessary piece to make gauge-invariantthe Lagrangian term L q = − X i s i m i ( ¯ ψ i + ig s i β i µ PV e φ a ¯ ψT a )( ψ i − ig s i β i µ PV e φ a T a ψ ) , (21)which is to replace the quark-mass term − P i s i m ¯ ψ i ψ i in (13). Thus, each PV quark can have a different mass m i , with m the mass of the physical quark.This new term can be written as L q = − X i s i m i ¯ ψ i ψ i − ig m PV µ PV h ¯ ψT a e φ a e ψ − ¯ e ψT a e φ a ψ i − X i s i m i β i µ ¯ ψT a e φ a T b e φ b ψ, (22)where we define a PV-quark mass scale m PV ≡ max i m i and a new null combination of quark fields e ψ = X i β i m i m PV ψ i . (23)For this to be null, there is the additional constraint X i s i m i β i = 0 , (24)and for e ψ and ψ to be mutually null, we must have X i s i m i β i = 0 . (25)This last constraint has the advantage of making thelast term in (22) simply zero. Also, the second andthird terms combined are proportional to P ij β i β j ( m j − m i ) ¯ ψ i T a e φ a ψ j , which is zero for i = j and guaranteesthat none of the scalars are directly coupled to the phys-ical quarks, as claimed above. There are now three con-straints on the PV quark masses and coupling coeffi-cients, which requires three PV quarks if all masses m i are to be chosen independently.The last step of the construction is the Faddeev–Popovghost term, which can be done by the usual methods ofpath-integral quantization [21]. We obtain, for ghosts c ak and anti-ghosts ¯ c ak , L FP = X k r k ∂ µ ¯ c ak ∂ µ c ak − X k r k µ k ζ ¯ c ak c ak (26)+ gf abc (cid:20) ∂ µ ¯ c a c b A µc − µ ζ ¯ e c a Z x dx ′ µ c b ( x ′ ) A µc ( x ′ ) (cid:21) , with the null combinations defined as c a ≡ X k ξ k c ak , ¯ c a ≡ X k ξ k ¯ c ak , ¯ e c a ≡ X k ξ k µ k µ ¯ c ak . (27)For these to be (mutually) null, we require X k r k µ k ξ k = 0 , X k r k µ k ξ k = 0 . (28)There are now four constraints on the PV-gluon masses µ k and coupling coefficients ξ k : the two above plus (6)and (19). If all the PV-gluon masses are to be chosenindependently, this would require four PV gluons; forconstrained mass values, the number of PV gluons canbe less. The number of PV ghosts is the same as the number of PV gluons; the number of PV scalars is onemore, except for pure Yang–Mills theory, where φ a isnot needed. Because µ ≪ µ PV , there could be difficultywith significant figures in numerical solutions of theseconstraints. However, calculations in QED [8] have indi-cated that the µ /µ PV ratio need not be extremely small,so that this potential difficulty may be mitigated.This completes the construction. The full expressionfor the PV-regulated QCD Lagrangian can be obtainedby adding L g and L FP , found in (17) and (26), to L in(13), and replacing the quark mass term in L with L q ,found in (22).The associated BRST transformations are determinedfrom the gauge transformations with the replacementΛ ak → ǫc ak , where ǫ is a real Grassmann constant forwhich ǫ = 0. The transformations are δA µak = ǫ∂ µ c ak + ǫr k ξ k gf abc c b A µc , (29) δψ i = iǫgs i β i T a c a ψ, δ ¯ ψ i = − iǫgs i β i ¯ ψT a c a , (30) δφ ak = ǫµ k c ak (31)+ ǫr k ξ k µ k gf abc Z x dx ′ µ c b ( x ′ ) A µc ( x ′ ) ,δ∂ µ φ ak = ǫµ k ∂ µ c ak + ǫr k ξ k µ k gf abc c b A µc , (32) δ ¯ c ak = − ζǫ (cid:18) ∂ µ A µak + µ k ζ φ ak (cid:19) , (33) δc ak = 12 ǫr k ξ k gf abc c b c c . (34)For the various null combinations we then find δA µa = ǫ∂ µ c a , δ e φ a = ǫµ PV c a , δc a = 0 , (35) δψ = 0 , δ e ψ = 0 . (36)All of the gauge-invariant pieces of the Lagrangian arethen automatically BRST invariant, and we need onlycheck the sum of the second term of L g and L FP , in (16)and (26) respectively. A short calculation shows that thissum, and therefore the entire Lagrangian, is also BRSTinvariant.However, what we have done is only a formal construc-tion. A next step is to construct the corresponding light-front Fock-space Hamiltonian and then to solve for itseigenstates. This will be facilitated by another aspect ofnull couplings, in that the projections [4] ψ − ≡ γ γ − ψ and e ψ − ≡ γ γ − e ψ of the Dirac field are gauge invari-ant and satisfy constraint equations, projected from theDirac equation, that are independent of the gluon fields.This allows for explicit solution of the constraint with-out invocation of light-cone gauge [6]. There will also becancellation of instantaneous fermion interactions, justas seen in Yukawa theory [12] and QED [6]. 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